This dissertation consists of two distinct parts, the first concerning S^1-equivariant cohomology of loop spaces and the second concerning stability in partial differential equations. In the first part of this dissertation, we study the existence of S^1-equivariant characteristic classes on certain natural infinite rank bundles over the loop space LM of a manifold M. We discuss the different S^1-equivariant cohomology theories in the literature and clarify their relationships. We attempt to use S^1-equivariant Chern-Weil techniques to construct S^1-equivariant characteristic classes. The main result is the construction of a sequence of S^1-equivariant characteristic classes on the total space of the bundles, but these classes do not descend to the base LM. In addition, we identify a class of bundles for which a single S^1-equivariant characteristic class does admit an S^1-equivariant Chern-Weil construction. In the second part of this dissertation, we study the Maslov index as a tool to analyze stability of steady state solutions to a reaction-diffusion equation in one spatial dimension. We show that the path of unstable subspaces associated to this equation is governed by a matrix Riccati equation whose solution S develops singularities when changes in the Maslov index occur. Our main result proves that at these singularities the change in Maslov index equals the number of eigenvalues of S that increase to +∞ minus the number of eigenvalues that decrease to -∞.

京都大学 / 0048 / 新制・課程博士 / 博士(理学) / 甲第21531号 / 理博第4438号 / 新制||理||1638(附属図書館) / 京都大学大学院理学研究科数学・数理解析専攻 / (主査)准教授 COLLINS Benoit Vincent Pierre, 教授 堤 誉志雄, 教授 泉 正己 / 学位規則第4条第1項該当 / Doctor of Science / Kyoto University / DFAM

The construction of characteristic classes via the curvature form of a connection is one motivation for the refinement of integral cohomology by de Rham cocycles -- known as differential cohomology. We will discuss the analog in the case of a group action on the manifold: We will show the compatibility of the equivariant characteristic class in integral Borel cohomology with the equivariant characteristic form in the Cartan model. Motivated by this understanding of characteristic forms, we define equivariant differential cohomology as a refinement of equivariant integral cohomology by Cartan cocycles.

Im Mittelpunkt der Untersuchungen stehen Grundlagen für äquivariante motivische Homotopietheorie. Für eine neue Grothendieck-Topologie auf einer Kategorie von äquivarianten glatten k-Schemata werden unstabile und stabile motivische Homotopietheorie entwickelt. Im zweiten Teil der Arbeit wird als Anwendung der stabilen Theorie eine Adams-Spektralsequenz mit motivischer Borel-Kohomologie konstruiert.

In this thesis we study symmetries of quantum field theory visible only at the non-perturbative level, which arise from large deformations of the integration contour in the path integral. We exposit the recently-developed theory of qq-characters that organizes such symmetries in the case of N = 2 supersymmetric gauge theories in four dimensions. We sketch the physical origin of such observables from intersecting branes in string theory, and the mathematical origin as certainequivariant integrals over Nakajima quiver varieties. We explain some of the main applications, including the derivation of Seiberg-Witten geometry for quiver gauge theories and the relations to quantum integrable systems.

<p>Isotropy representations provide powerful tools for understanding the classification of equivariant principal bundles over the $2$-sphere. We consider a $\Gamma$-equivariant principal $G$-bundle over $S^2$ with structural group $G$ a compact connected Lie group, and $\Gamma \subset SO(3)$ a finite group acting linearly on $S^2.$ Let $X$ be a topological space and $\Gamma$ be a group acting on $X.$ An isotropy subgroup is defined by $\Gamma_x = \{\gamma \in \Gamma \lvert \gamma x=x\}.$ Assume $X$ is a $\Gamma$-space and $A$ is the orbit space of $X$. Let $\varphi: A\rightarrow X$ be a continuous map with $\pi \circ \varphi = 1_A$. An isotropy groupoid is defined by $\mathfrak{I} = \{(\gamma,a) \in \Gamma\times A \lvert \ \gamma \in \Gamma_{\varphi(a)}\}.$ An isotropy representation of $\mathfrak{I}$ is a continuous map $\iota : \mathfrak{I} \rightarrow G$ such that the restriction map $\mathfrak{I}_a \rightarrow G$ is a group homomorphism. $\Gamma$- equivariant principal $G$-bundles are studied in two steps; \begin{enumerate} [1)] \item the restriction of an equivariant bundle to the $\Gamma$ equivariant 1-skeleton $X \subset S^2$ where $\mathfrak{I}$ is isotropy representation of $X$ over singular set of the $\Gamma$-sets in $S^2$ \item the underlying $G$-bundle $\xi$ over $S^2$ determined by $c(\xi)\in \pi_2(BG).$ \end{enumerate}</p> / Master of Science (MSc)

The focus of this thesis is the dynamical behaviour of linear arrays of laser oscillators with nearest-neighbour coupling. In particular, we study how laser dynamics are influenced by laser-coupling strength, $\kappa$, the natural frequencies of the uncoupled lasers, $\tilde{\Omega}_j$, and the coupling between the magnitude and phase of each lasers electric field, $\alpha$. Equivariant bifurcation analysis, combined with Lyapunov exponent calculations, is used to study different aspects of the laser dynamics. Firstly, codimension-one and -two bifurcations of relative equilibria determine the laser coupling conditions required to achieve stable phase locking. Furthermore, we find that global bifurcations and their associated infinite cascades of local bifurcations are responsible for interesting locking-unlocking transitions. Secondly, for large $\alpha$, vast regions of the parameter space are found to support chaotic dynamics. We explain this phenomenon through simulations of $\alpha$-induced stretching-and-folding of the phase space that is responsible for the creation of horseshoes. A comparison between the results of a simple {\it coupled-laser model} and a more accurate {\it composite-cavity mode model} reveals a good agreement, which further supports the use of the simpler model to study coupling-induced instabilities in laser arrays. Finally, synchronisation properties of the laser array are studied. Laser coupling conditions are derived that guarantee the existence of synchronised solutions where all the lasers emit light with the same frequency and intensity. Analytical stability conditions are obtained for two special cases of such laser synchronisation: (i) where all the lasers oscillate in-phase with each other and (ii) where each laser oscillates in anti-phase with its direct neighbours. Transitions from complete synchronisation (where all the lasers synchronise) to optical turbulence (where no lasers synchronise and each laser is chaotic in time) are studied and explained through symmetry breaking bifurcations. Lastly, the effect of increasing the number of lasers in the array is discussed in relation to persistent optical turbulence.

No presente trabalho construímos novos exemplos de hipersuperfícies mínimas, completas e H-equivariantes de espaços simétricos. Para tal, usamos o método da geometria diferencial equivariante (Hsiang-Lawson). Dividimos nosso estudo em duas partes, a saber, espaços simétricos G/K de tipo não compacto e compacto. No primeiro caso são estudadas ações polares de subgrupos H adaptados à decomposição de Iwasawa G=KAN. No segundo caso usamos a classificação (Podesta-Thobergsson) dos subgrupos H de Spin(9) que atuam com cohomogeneidade dois sobre o plano projetivo octoniônico F_4/Spin(9). / In the present work we construct new examples of complete minimal H-equivariant hypersurfaces of symmetric spaces G/K. For that, we use the equivariant differential geometry method (Hsiang-Lawson). We divide our research in two parts, namely, symmetric spaces of non-compact and compact type. In the first case we study polar actions of subgroups H adapted to the Iwasawa decomposition G=KAN. In the second case we use the classification (Podesta-Thobergsson) of the subgroups H of Spin(9) which act with cohomogeneity two on the octonionc projective plane F_4/Spin(9).

In this thesis, we investigate a special class of multi-agent systems, which we call reciprocal multi-agent (RMA) systems. The evolution of agents in a RMA system is governed by interactions between pairs of agents. Each interaction is reciprocal, and the magnitude of attraction/repulsion depends only on distances between agents. We investigate the class of RMA systems from four perspectives, these are two basic properties of the dynamical system, one formula for computing the Morse indices/co-indices of critical formations, and one formation control model as a variation of the class of RMA systems. An important aspect about RMA systems is that there is an equivariant potential function associated with each RMA system so that the equations of motion of agents are actually a gradient flow. The two basic properties about this class of gradient systems we will investigate are about the convergence of the gradient flow, and about the question whether the associated potential function is generically an equivariant Morse function. We develop systematic approaches for studying these two problems, and establish important results. A RMA system often has multiple critical formations and in general, these are hard to locate. So in this thesis, we consider a special class of RMA systems whereby there is a geometric characterization for each critical formation. A formula associated with the characterization is developed for computing the Morse index/co-index of each critical formation. This formula has a potential impact on the design and control of RMA systems. In this thesis, we also consider a formation control model whereby the control of formation is achieved by varying interactions between selected pairs of agents. This model can be interpreted in different ways in terms of patterns of information flow, and we establish results about the controllability of this control system for both centralized and decentralized problems. / Engineering and Applied Sciences

A state sum is an expression approximating the partition function of a d-dimensional field theory on a closed d-manifold from a triangulation of that manifold. To consider state sums in completely local 2-dimensional topological field theories (TFT's), we introduce a mechanism for incorporating triangulations of surfaces into the cobordism ([infinity],2)-category. This serves to produce a state sum formula for any fully extended 2-dimensional TFT possibly with extra structure. We then follow the Cobordism Hypothesis in classifying fully extended 2-dimensional G-equivariant TFT's for a finite group G. These are oriented theories in which bordisms are equipped with principal G-bundles. Combining the mechanism mentioned above with our classification results, we derive Turaev's state sum formula for such theories. / text